关于n×n复矩阵与其诱导矩阵之间的关系

The Relation between n × n Complex Matrix and Its Induced Matrix

讨论三个问题:a.设A是n×n复矩阵,且K(A)分别是正规的、厄米特的、半正定的和反厄米特的,用简洁的方法证明A的某些性质;b.设A是复可逆矩阵,巨C_m(A)分别是正规的、厄米特的、正定的和反厄米特的,讨论A具有的性质的条件;c.设A,B均为n×n复矩阵,讨论C_m(A)=C_m(B)的必要充分条件.

This paper deals with three problems: (1) Let A be an n×n complex matrix and an induced matrix K(A) be normal, Hermitian, semi-positive definite and skew-Hermitian. Some properties of A are tersely proved. (2) Let A be a complex nonsingular matrix and an induced matrix Cm(A) be normal, Hermitian, positive definite and skew-Hermitian. The conditions under which A has these properties are discussed. (3) Let A, B be n×n complex matrices. The necessary and sufficient conditions under which the relation Cm(A)=Cm(B) holds are discussed.